Question: Determine how many solutions exist for the system of equations. ${6x-y = -3}$ ${-18x+3y = 9}$
Convert both equations to slope-intercept form: ${6x-y = -3}$ $6x{-6x} - y = -3{-6x}$ $-y = -3-6x$ $y = 3+6x$ ${y = 6x+3}$ ${-18x+3y = 9}$ $-18x{+18x} + 3y = 9{+18x}$ $3y = 9+18x$ $y = 3+6x$ ${y = 6x+3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x+3}$ ${y = 6x+3}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${6x-y = -3}$ is also a solution of ${-18x+3y = 9}$, there are infinitely many solutions.